Hal writes:
>It is an interesting thought that an earlier generation of replicators
>burned through some currently unavailable energy source and produced the
>dark matter. They neglected stars because either they didn't exist yet
>or their output was too paltry to be concerned with. ...
>Carl's reasoning emphasizes the behavior of replicators at the edges
>of the sphere. If their whole point is speed, they might well ignore
>resources which take more time to develop. I think Robin's equations
>will show the same thing. You have to look behind the frontier though and
>try to understand why there were no low-budget colonies left behind to
>use the remaining resources. It's possible that there was too much of a
>parameter-space gap to evolve from the fast-moving replicators to the
>kind which could use stars and other low energy sources. In a sense,
>we may be the first replicators able to utilize this source of energy.

Let R(T) be the resources available at an oasis after growing there
for time T. Assume ln(R(T)) is increasing and concave in T with R(0) = 1.
This models Carl's observation that some resources at an oasis allow
fast growth, and others require more patience to exploit.

As before, assume a cost C > 1 to make a probe which may eventually
deliver a resource of 1 to a new oasis, a delay D to speed up and slow
down, a constant velocity V while traveling, a chance of A per unit
distance of being destroyed, and a chance Q that an oasis is suitable.

The maximum sustainable speed S is then given by

1 1 (D + T)
- = - + -------------- ,
S V A*ln(Q*R(T)/C)

and no probes leave an oasis until some optimal T*, at which point
all of R(T*) is used to make probes. The optimal T* is the tangency
point where a line from the point (-D,-ln(Q/C)) touches the ln(R(T))
curve.

Note that if a probe falls behind the front of the colonization wave,
it faces a smaller chance of finding a suitable oasis not already
occupied by a previous probe. Thus Q declines behind the frontier.
Thus the max sustainable speed is less behind the front, so the
"depth" of the front region should spread out with time. Furthermore,
since T* increases as Q declines, selection seems to favor probes who
use up more local resources R(T*) before moving on. As Q approaches
zero, T* grows until sustained colonization becomes impossible, at
Qmin, where C(V) >= Qmin*R(infinity) for any V.

Thus it seems replicators won't leave much resources untouched.
It seems that any earlier replicators must have been stupid
in this sense: they couldn't adjust their strategy to crowding,
or they just couldn't make replicators out of the resources we use.